factor by grouping

Question

25u³+110u²-75u

 

Factor by grouping. I don't understand this one. I just can't figure it out. 

Answer 1

Alright, Noelle, this is a different way to factor.

 

The first thing we need to do is factor out anything in common. 5u is common among all three terms, so our equation now looks like 5u(5u² + 22u - 15). Got it?

 

What we need to do next is multiply the first and third coefficients.  5 x -15 = -75.

 

Now, we need to find two factors of -75 that equal the middle coefficent, 22, when they are added together. Remember that one factor is going to have to be negative in order to make 75 negative, so we need to pick two factors that are 22 apart.

 

25 and 3 are 22 apart, and 25 x 3 = 75. Since 22 is positive in our equation, then we have 25 and -3.

 

We now split the 22u into our two factors.

 

5u(5u² + 25u - 3u - 15). It does NOT matter if 25u or -3u is first.

 

Now, what is common between 5u² and 25u? 5u, right?

 

What is in common between -3u and -15. -3. Got it?

 

We factor both out.

 

5u(5u(u + 5) -3(u + 5))

 

Assuming we did this right, what is inside both sets of parentheses is the same! We can combine what is in front of them into an extra set of parentheses.

 

5u((5u -3)(u + 5)).  There is our answer.

 

As a simple example, let's take 5(100) - 3(100).

 

We would normally say 500 - 300 = 200, but (5 - 3)(100) = 2(100) = 200, also.

 

Hope this helps!

 

 

 

 

Answer 2

Hi Noelle

 

First we need to factor out a u since all three terms have that in common.

 

u (25u² + 110u - 75).

 

Next we need to divide each term by the g.c.f (greatest common factor).

 

In this particular case, the greatest common factor is 5.

 

Once we divide each term by 5 we get...

 

u (5u² + 22u - 15)

 

Now we un-foil, or factor the polynomial even further to get two binomials.

 

u (5u - 3)(u + 5)

 

Our solutions are now...

 

I) 5u - 3 = 0

II) u + 5 = 0

 

First gives us u = 3/5

Second gives us u = -5

Third gives us u = 0 (this is from the u that we initially factored out)

 

We can check by substituting both 3/5 and -5 in for x in the original equation.

 

Substituting 3/5 for x gives us 0.

Substituting -5 for x gives us 0.

 

Substituting 0 for x gives us 0.

 

Therefore, our solution set is {-5, 0, 3/5} for x.